Zimmer’s conjecture for actions of
Abstract.
We prove Zimmer’s conjecture for actions by finite-index subgroups of provided . The method utilizes many ingredients from our earlier proof of the conjecture for actions by cocompact lattices in [BFH] but new ideas are needed to overcome the lack of compactness of the space (admitting the induced -action). Non-compactness allows both measures and Lyapunov exponents to escape to infinity under averaging and a number of algebraic, geometric, and dynamical tools are used control this escape. New ideas are provided by the work of Lubotzky, Mozes, and Raghunathan on the structure of nonuniform lattices and, in particular, of providing a geometric decomposition of the cusp into rank one directions, whose geometry is more easily controlled. The proof also makes use of a precise quantitative form of non-divergence of unipotent orbits by Kleinbock and Margulis, and an extension by de la Salle of strong property (T) to representations of nonuniform lattices.
1. Introduction
1.1. Statement of results
The main result of this paper is the following:
Theorem A.
Let be a finite-index subgroup of and let be a closed manifold of dimension . If is a group homomorphism then is finite111After this work was completed, Brown-Damjanovic-Zhang showed that some modifications of our arguments also give a proof for diffeomorphisms [BDZ].. In addition, if is a volume form on , and if , then if and is a group homomorphism then is finite.
For , we remark that the conclusion of Theorem A is known for actions on the circle by results of Witte Morris [Wit] (see also [Ghy, BM] for actions by more general lattices on the circle) and for volume-preserving actions on surfaces by results of Franks and Handel and of Polterovich [FH, Pol]. The proof in this paper requires that though we expect it can be modified to cover actions by ; since these results are not new, we only present the case for . While this is a very special case of Zimmer’s conjecture, it is a key example. For instance, the version of Zimmer’s conjecture restated by Margulis in his problem list [Mar2] is a special case of Theorem A.
Note that if is a finite-index subgroup of acting on compact manifold , we may induce an action of on a (possibly non-connected) compact manifold where if there is with and . Connectedness of is neither assumed nor is it used in either the proof of Theorem A or in [BFH]. Thus, for the remainder we will simply assume .
This paper is a first step in extending the results in [BFH] to the case where is a nonuniform lattice in a split simple Lie group and the strategy of the proof of Theorem A relies strongly on the strategy used in [BFH]. In the remainder of the introduction, we recall the proof in the cocompact case, indicate where the difficulties arise in the nonuniform case, and outline the proof of Theorem A. At the end of the introduction we make some remarks on other approaches and difficulties we encountered.
We recall a key definition from [BFH]. Let be a finitely generated group. Let denote the word-length function with respect to some choice of finite generating set for . Given a diffeomorphism let (for some choice of norm on ).
Definition 1.1.
An action has uniform subexponential growth of derivatives if
| for every , there is such that for all | (1) |
The main result of the paper is the following:
Theorem B.
For , let and let be a closed manifold.
-
(1)
If then any action has uniform subexponential growth of derivatives;
-
(2)
if is a volume form on and then any action has uniform subexponential growth of derivatives.
To deduce Theorem A from Theorem B, we apply [BFH, Theorem 2.9] and de la Salle’s recent result establishing strong property for nonuniform lattices [dlS, Theorem 1.2] and conclude that any action as in Theorem A preserves a continuous Riemannian metric. For clarity, we point out that we need de la Salle’s Theorem and not his Theorem because we need the measures converging to the projection to be positive measures. That Theorem [dlS, Theorem 1.2] provides positive measures where [dlS, Theorem 1.1] does not is further clarified in [dlS, Section 2.3]. Once a continuous invariant metric is preserved, the image of any homomorphism in Theorem A is contained in a compact Lie group . All such homomorphisms necessarily have finite image due to the presence of unipotent elements in . We remark that while the finiteness of the image of was deduced using Margulis’s superrigidity theorem in [BFH], it is unnecessary in the setting of Theorem A since, as any unipotent element of lies in the center of some integral Heisenberg subgroup of , all unipotent elements have finite image in and therefore so does .
1.2. Review of the cocompact case
To explain the proof of Theorem B, we briefly explain the difficulties in extending the arguments from [BFH] to the setting of actions by nonuniform lattices. We begin by recalling the proof in the cocompact setting.
In both [BFH] and the proof of Theorem B, we consider a fiber bundle
which allows us to replace the -action on with a -action on . In the case that is cocompact, showing subexponential growth of derivatives of the -action is equivalent to showing subexponential growth of the fiberwise derivative cocycle for the -action.
To prove such subexponential growth for the -action on we argued by contradiction to obtain a sequence of points and semisimple elements in a Cartan subgroup which satisfy for some . Here denotes the derivative of translation by at , is the fiberwise tangent bundle of , and is the restriction of to .
The pairs determine empirical measures on supported on the orbit which accumulate on a measure that is -invariant for some and has a positive Lyapunov exponent for the fiberwise derivative cocycle of size at least . Using classical results in homogeneous dynamics in conjunction with the key proposition from [BRHW], we averaged the measure to obtain a -invariant measure on with a non-zero fiberwise Lyapunov exponent; the existence of such a measure contradicts Zimmer’s cocycle superrigidity theorem.
1.3. Difficulties in the nonuniform setting.
When is nonuniform the space is not compact and the sequence of empirical measures might diverge to infinity in ; that is, in the limit we might have a “loss of mass”. Additionally, even if the measures satisfy some tightness criteria so as to prevent escape of mass, one might have “escape of Lyapunov exponents:” for a limiting measure , the Lyapunov exponents may be infinite or the value could drop below the value expected by the growth of fiberwise cocycles along the orbits . For instance, the contribution to the exponential growth of derivatives along the sequence of empirical measures could arise primarily from excursions of orbits deep into the cusp. If one makes naïve computations with the return cocycle (measuring for in a fundamental domain the element of needed to bring back to a ) one in fact expects that the fiberwise derivative are very large for translations of points far out in the cusp since the orbits of such points cross a large number of fundamental domains. The weakest consequence of this observation is that subexponential growth of the fiberwise derivative of the induced -action is much stronger than subexponential growth of derivatives of the -action. While we still work with the induced -action and the fiberwise derivative in many places, the arguments become more complicated than in the cocompact case.
In the homogeneous dynamics literature, there are many tools to study escape of mass. Controlling the escape of Lyapunov exponents seems to be more novel. To rule out escape of mass, it suffices to prove tightness of family of measures To control Lyapunov exponents, we introduce a quantitative tightness condition: we construct measures with uniformly exponentially small mass in the cusps. See Section 3.2. It is a standard computation to show the Haar measure on (or any where is semisimple and is a lattice) has exponentially small mass in the cusps.
1.4. Outline of proof
With the above difficulties in mind, we outline the strategy of the proof of Theorem B. Lubotzky, Mozes and Raghunathan proved that is quasi-isometrically embedded in . And in this special case, they give a proof that every element can be written as a product of at most elements contained in canonical copies of determined by pairs of standard basis vectors for ; moreover the word-length of each is at most proportional to the word-length of [LMR1, Corollary 3]. (We note however that such effective generation of only holds for ; for the general case, in [LMR2] a weaker generation of in terms of -rank 1 subgroups is shown.) Thus, to show uniform subexponential growth of derivatives for the action of , it suffices to show uniform subexponential growth of derivatives for the restriction of our action to each canonical copy of .
We first obtain uniform subexponential growth of derivatives for the unipotent elements in in Section 4. See Proposition 4.1. The strategy is to consider a subgroup of the form . We first prove that a large proportion of elements in satisfy (1). To prove this, we use that if then a typical -orbit in equidistributes to the Haar measure. In particular, for the empirical measures along such -orbits we apply the techniques from [BFH] to show subexponential growth of fiberwise derivatives along such orbits and conclude that a large proportion of satisfies (1). See Proposition 4.2. The proof of this fact repeats most of the ideas and techniques from [BFH] as well a quantitative non-divergence of unipotent averages following Kleinbock and Margulis. The precise averaging procedure is different here than in [BFH].
Having shown Proposition 4.2, we consider the -action on the normal subgroup of to show that for every , the ball of radius in contains a positive-density subset of unipotent elements satisfying (1). Taking iterated sumsets of such good unipotent elements of with a finite set one obtains uniform subexponential growth of derivatives for every element in . This relies heavily on the fact that is abelian. See Subsection 4.2.
It is worth noting that the subgroups of the form are also considered in the work of Lubotzky, Mozes, and Raghunathan in [LMR1] as well as in Margulis’s early constructions of expander graphs and subsequent work on property (T) and expanders [Mar1].
Having established Proposition 4.1, we assume for the sake of contradiction that the restriction of to fails to exhibit uniform subexponential growth of derivatives. We obtain in Subsection 5.2 a sequence of -orbit segments in which drift only a sub-linear distance into the cusp with respect to their length and accumulate exponential growth of the fiberwise derivative. Here we use that orbits deep in the cusp of correspond to unipotent deck transformations and that Proposition 4.1 implies that these do not contribute to the exponential growth of the fiberwise derivative. Here, we heavily use the structure of subgroups.
We promote the family of orbit segments in to a family of measures all of whose subsequential limits are -invariant measures on with non-zero fiberwise exponents. To construct , we construct a Følner sequence inside a solvable subgroup where is the full Cartan subgroup of and is a well-chosen abelian subgroup of unipotent elements. We average our orbit segments over to obtain the sequence of measures in . In general, Følner sets for are subsets which are linearly large in the -direction and exponentially large in the direction. In our case the -part will not affect the Lyapunov exponent because we work inside a subset where the return cocycle restricted to takes unipotent values and we have already proven subexponential growth of the fiberwise derivatives for unipotent elements.
The fact that behaves well in the cusp is due to two facts: First, the segments obtained in Subsection 5.2 do not drift too deep into the cusp of . Second, we choose our subgroup such that the -orbits of each point along each is a closed torus that is well-behaved when translated by . The argument here is related to the fact closed horocycles in the cusp of equidistribute to the Haar measure when flowed backwards by the geodesic flow.
To finish the argument, we show that any -invariant measure on projects to Haar measure on using Ratner’s measure classification and equidistribution theorems. Then, as in [BFH], we can use [BRHW, Proposition 5.1] and argue as in the cocompact case in [BFH] show that is in fact -invariant and thereby obtain a contradiction with Zimmer’s cocycle superrigidity theorem.
1.5. A few remarks on other approaches.
We close the introduction by making some remarks on other approaches, particularly other approaches for controlling the escape of mass. We emphasize here that one key difficulty for all approaches is that we are not able to control the “images” of the cocycle in either our special case or in general. To understand this remark better, consider first the case where and . If we take a one-parameter subgroup and take the trajectory for in some interval and assume and assume the entire trajectory on lies deep enough in the cusp, then is necessarily unipotent for all in . No similar statement is true for and . In fact analogous statements are true if and only if has -rank one, this is closely related to the fact that higher -rank locally symmetric spaces are -connected at infinity. This forces us to “factor” the action into actions of rank-one subgroups in order to control the growth of derivatives.
One might hope to obtain subexponential growth of derivatives more directly for all elements of , or even directly in , by proving better estimates on the size of the “generic” subsets of (or ) whose -orbits define empirical measures satisfying some tightness condition. While one can get good estimates on the size of the sets in Proposition 4.2 using Margulis functions and large deviation estimates as in [Ath, EM2], the resulting estimates are not sharp enough to allow us to prove subexponential growth of derivatives. One can compare with the conjectures in [KKLM] about loss of mass.
An elementary related question is the following: Let be a ball of radius in a Lie group (or a lattice ) and suppose there exists subset of such that and have more or less equal mass, meaning that:
for a certain sequence of numbers converging to zero. Does there exists an integer (independent of ) such that for large:
| (2) |
Observe that the question depends on how fast is decreasing and on the group . For example if abelian, can be a sufficiently small constant as a consequence of Proposition 4.9. Also, it is not hard to see that for any group the existence of is guaranteed if decreases exponentially quickly. So the real question is how fast has to decrease to zero in order for this statement to hold. Does (2) holds for and for some ? If the answer to this question is yes, then it would be possible to approach our results via Margulis functions and large deviation estimates.
Acknowledgements
We thank Dave Witte Morris for his generous willingness to answer questions of all sorts throughout the production of this paper and [BFH]. We also thank to Shirali Kadyrov, Jayadev Athreya and Alex Eskin for helpful conversations, particularly on the material in Subsection 1.5 and Mikael de la Salle for many helpful conversations regarding strong property . We also thank the anonymous referee for a very careful reading and numerous comments which helped improve the exposition.
2. Standing notation
We review the notation introduced in [BFH] and establish some standing notation and conventions as well as state some facts used in the remainder of the paper.
2.1. Lie theoretic and geometric notation
We write and . Let denote the Lie algebra of . Let denote the identity element of . We fix the standard Cartan involution given by and write and , respectively, for the and eigenspaces of . Define to be a maximal abelian subalgebra of . Then is the vector space of diagonal matrices.
The roots of are the linear functionals defined as
The simple positive roots are and the positive roots are the positive integral combinations of that are still roots.
For a root , write for the associated root space. Each root space exponentiates to a 1-parameter unipotent subgroup . The Lie subalgebra generated by all root spaces for positive roots , coincides with the Lie algebra of all strictly upper-triangular matrices.
Let and be the analytic subgroups of corresponding to and . Then
-
(1)
is the group of all diagonal matrices with positive entries. is an abelian group and we identity linear functionals on with linear functionals on via the exponential map ;
-
(2)
is the group of upper-triangular matrices with s on the diagonal;
-
(3)
.
The Weyl group of is the group of permutation matrices. This acts transitively on the set of all roots .
For , the subgroup of generated by and is isomorphic to . We denote this subgroup by Then is a lattice in isomorphic to . Note then that is the unit tangent bundle to the modular surface. We will use the standard notation for an elementary matrix with 1s on the diagonal and in the -place and 0s everywhere else. Note that and generate .
We equip with a left--invariant and right--invariant metric. Such a metric is unique up to scaling. Let denote be the induced distance on . With respect to this metric and distance , each is geodesically embedded. By rescaling the metric, we may assume the restriction of to each coincides with the standard metric of constant curvature on the upper half plane . This metric has the following properties that we exploit throughout.
-
(1)
For any matrix norm on there is a such that
(3) for all .
-
(2)
Let denote the metric ball of radius in centered at . Then with respect to the induced Riemannian volume on we have
and for all sufficiently large
(4) -
(3)
For any matrix norm on , there are constants and such that for any matrix we have
(5) -
(4)
In particular, there are and so that if is an elementary unipotent matrix then
(6)
2.2. Suspension space and induced -action
Let be the fiber-bundle over obtained as follows: on let act as
and let act as
The -action on descends to a -action on the quotient . Let be the canonical projection. As in [BFH], we write for the fiberwise tangent bundle to . Write for the projectivization of the fiberwise tangent bundle. We write for the fiberwise derivative as in [BFH]. For and , write
for the action of on induced by .
We follow [BRHW, Section 2.1] and equip with a Riemannian metric with the following properties:
-
(1)
is -invariant.
-
(2)
for and , under the canonical identification of the -orbit of with , the restriction of to the -orbit of coincides with the fixed right-invariant metric on .
-
(3)
There is a Siegel fundamental set and such that for any , the map distorts the restrictions of to and by at most .
The metric then descends to a Riemannian metric on . Note that by averaging the metric over the left action of , we may also assume that the metric on is left--invariant. This, in particular, implies the right-invariant metric on in above is chosen to be left--invariant.
To analyze the coarse dynamics of the suspension action, it is often useful to consider the return cocycle . This cocycle is defined relative to a fundamental domain for the right -action on . For any , take to be the unique lift of in and define to be the unique element of such that . Any two choices of fundamental domain for define cohomologous cocycles but we require a choice of well-controlled fundamental domains . Namely, we choose to either be contained in a Siegel fundamental set or to be a Dirichlet domain for the identity. With these choices, we have the following.
Let denote the Dirichlet domain of the identity for the action on ; that is
Since each is geodesically embedded in and since , it follows
| (7) |
is a Dirichlet domain of the identity for the -action on . Viewing acting on the upper half-plane model of hyperbolic space by Möbius transformations is the standard Dirichlet domain for the modular surface, the hyperbolic triangle with endpoints at , , and .
Lemma 2.1.
If is either contained in either a Siegel fundamental set or a Dirichlet domain for the identity then there is a constant such that for all and
In the above lemma, is the word-length of , is the distance from to in , and is the distance from to the identity coset in . For a Dirichlet domain for the identity, the Lemma is shown in [Sha2, §2]; for fundamental domains contained in Siegel fundamental sets, the estimate follows from [FM, Corollary 3.19] and the fact that the distance to the identity in a Siegel domain is quasi-Lipschitz equivalent to the distance to the identity in the quotient . Both estimates heavily use the main theorem of Lubotzky, Mozes, and Raghunathan [LMR1, LMR2] to compare the word-length of with .
Fix once and for all a fundamental domain .
The estimates in Lemma 2.1 is often used to obtain integrability properties of and related cocycles with respect to the Haar measure on . As the function is in for any compact set we have that
is in for all . In the sequel, we typically do not directly use the integrability properties (since we work with measures other than Haar) but rather the estimate in Lemma 2.1.
3. Preliminaries on measures, averaging, and Lyapunov exponents
We present a number of technical facts regarding invariant measures, equidistribution, averaging, and Lyapunov exponents that will be used in the remainder of the paper.
3.1. Ratner’s measure classification and equidistribution theorems
We recall Ratner’s theorems on equidistribution of unipotent flows. Let be a 1-parameter unipotent subgroup in . Given any Borel probability measure on let
Theorem 3.1 (Ratner).
Let be a 1-parameter unipotent subgroup and consider the action on . The following hold:
-
(a)
Every ergodic, -invariant probability measure on is homogeneous [Rat1, Theorem 1].
-
(b)
The orbit closure is homogeneous for every [Rat1, Theorem 3].
-
(c)
The orbit equidistributes in ; that is converges to the Haar measure on as .
-
(d)
Let be a root of and let be the Lie subalgebra generated by and . Let be an triple with and and let . Let .
Let be a -invariant Borel probability measure on . If is -invariant, then is -invariant.
Conclusion (d) follows from [Rat2, Proposition 2.1] and the structure of -triples. See also the discussion in the paragraph preceding [Rat1, Theorem 9]. In our earlier work on cocompact lattices [BFH], we averaged over higher-dimensional unipotent subgroups and required a variant of (c) due to Nimish Shah [Sha1]. Here we only average over one-dimensional root subgroups and can use the earlier version due to Ratner.
From Theorem 3.1, for any probability measure on it follows that the weak- limit
exists and that the -ergodic components of are homogeneous.
3.2. Measures with exponentially small mass in the cusps
We now define precisely the notion of measures with exponentially small mass in the cusps from the introduction. Let be a complete, second countable, metric space. Then is Polish. Let be a finite Borel (and hence Radon) measure on . We say that has exponentially small mass in the cusps with exponent if for all
| (8) |
for some (and hence any) choice of base point . We say that a collection of probability measures on has uniformly exponentially small mass in the cusps with exponent if for all
Below, we often work in in the setting where and and where the distance induced from a right-invariant metric on . When we interpret a point as a unimodular lattice . Fix any norm on and define the systole of a lattice to be
We have that
| (9) |
for some constants whence
Thus, if we only care about finding a positive exponent such that (8) holds for all , it suffices to find such that
| (10) |
We define the systolic exponent to be the supremum of all satisfying (10).
In the sequel, we will frequently use the following proposition to avoid escape of mass into the cusps of when averaging a measure along a unipotent flow.
Proposition 3.2.
Let be a 1-parameter unipotent subgroup of . Let be a probability measure on with exponentially small mass in the cusps. Then the family of measures
has uniformly exponentially small mass in the cusps.
3.3. Proof of Proposition 3.2
We first show that the family of averaged measures
has uniformly exponentially small mass in the cusps. The key idea is to use the quantitative non-divergence of unipotent orbits following Kleinbock and Margulis.
Lemma 3.3.
Let be a probability measure on with exponentially small mass in the cusps and systolic exponent .
Then the family of measures has uniformly exponentially small mass in the cusps with systolic exponent .
Proof.
Let be a discrete subgroup. Let denote the volume of where denotes the -span of . It follows from Minkowski’s lemma that there is a constant (depending only on ) such that if
then there is a non-zero vector with . In particular, if then for some constant we have
for all discrete subgroups .
From [KM, Theorem 5.3] as extended in [Kle, Theorem 0.1], there is a such that for every and , if then, since for every discrete subgroup , we have
| (11) |
where is the Lebesgue measure of the set . Note that (11) still holds even in the case . Note that if then for we have
In particular, when we have (for all including ) that
Then for and we have
which is uniformly bounded in as long as . ∎
For the limit measure we have the following which holds in full generality.
Lemma 3.4.
Let be a complete, second countable, metric space. Let be a sequence of Borel probability measures on converging in the weak- topology to a measure . If the family has uniformly exponentially small mass in the cusps with exponent then the limit has exponentially small mass in the cusps with exponent .
Proof.
We have that in the weak- topology. In particular, for any closed set and open set we have
Fix and take Fix with
for all . Using Markov’s inequality, for all and every we have
so
Then, for the limit measure , we have
3.4. Averaging certain measures on
Take to be the standard set of simple positive roots of :
Let be the analytic subgroup of whose Lie algebra is generated by roots spaces associated to and let be the analytic subgroup of whose Lie algebra is generated by roots spaces associated to . We have and . Then is the subgroup of all matrices of the form
where .
We let be the the co-rank-1 subgroup of the Cartan subgroup given by . Let be the highest positive root.
Proposition 3.5.
Let be any -invariant probability on . Let or and let or .
Then is -invariant and
is the Haar measure on .
Proof.
We have that is -invariant. Let and note that remains - and -invariant.
Case 1(a) : . Consider first the case that . Then remains invariant under and for all since these roots commute with . By Theorem 3.1(d) we have that is also invariant under and for all . Taking brackets, is invariant under for every positive root .
Case 1(b) : . Consider now the case that . Then remains invariant under and for all since these roots commute with . By Theorem 3.1(d) we have that is also invariant under and for all . Taking brackets, is invariant under for every positive root of the form for each . In particular, is invariant under and hence also invariant under . In particular is invariant under for every positive root .
Note that in either case, we have that is invariant under for every positive root .
Let .
Case 2(a) : . If , then remains invariant under and for all . Note additionally remains invariant under the highest-root group . Again, by Theorem 3.1(d) we have that is also invariant under and for all . In particular is also invariant under for every negative root . It follows as in Case 1(b) that is invariant under and hence invariant under for every positive root . Thus is -invariant.
Case 2(b) : . If , then remains invariant under and for all . Note additionally remains invariant under . Again, we have that is also invariant under and for all . In particular is also invariant under for every positive root . As in Case 1(b) that is invariant under and hence invariant under for every negative root . Thus is -invariant. ∎
3.5. Lyapunov exponents for unbounded cocycles
Let be a second countable, complete metric space. We moreover assume the metric is proper. Let act continuously on .
Let be a continuous, finite-dimensional vector bundle equipped with a norm . A linear cocycle over the -action on is an action by vector-bundle automorphisms that projects to the -action on . We write for the linear map between Banach spaces and . By the norm of we mean the operator norm and the conorm is . We say that is tempered with respect to the metric if there is a such that for any compact set and base point there is so that
and
where denotes the operator norm and denotes the operator conorm applied to linear maps between Banach spaces and .
If is a probability measure on with exponentially small mass in the cusps, it follows that the function is whence we immediately obtain the following.
Claim 3.6.
Let a probability measure on with exponentially small mass in the cusps. Suppose that is tempered. Then for any compact , the functions
are .
Given and an -invariant Borel probability measure on we define the average leading (or top) Lyapunov exponent of to be
| (12) |
From the integrability of the function we obtain the finiteness of Lyapunov exponents.
Corollary 3.7.
For and an -invariant probability measure on with exponentially small mass in the cusps, if is tempered then the average leading Lyapunov exponent of is finite.
Note that for an -invariant measure , the sequence is subadditive whence the infimum in (12) maybe replaced by a limit.
As in the case of bounded continuous linear cocycles, we obtain upper-semicontinuity of leading Lyapunov exponents for continuous tempered cocycles when restricted to families of measures with uniformly exponentially small measure in the cusp.
Lemma 3.8.
Let be a tempered cocycle. Given suppose the restriction of the cocycle to the action of is continuous.
Then—when restricted to a set of -invariant Borel probability measures with uniformly exponentially small mass in the cusps—the function
is upper-semicontinuous with respect to the weak- topology.
Proof.
Let be a family of -invariant Borel probability measures with uniformly exponentially small mass in the cusps. As the pointwise infimum of continuous functions is upper-semicontinuous, is enough to show that the function
is continuous with respect to the weak- topology for each . As the weak- topology is first countable, it is enough to show is sequentially continuous.
Let in . Given , fix a continuous with
As we assume our metric is proper, is a bounded continuous function whence
Moreover, there are and such that for all and
and
In particular,
Thus for any , we have
It follows that given there is so that
for all .
In particular, taking and sufficiently large we have
Sequential continuity then follows. ∎
3.6. Lyapunov exponents under averaging and limits
We now consider the behavior of the top Lyapunov exponent as we average an -invariant probability measure over an amenable subgroup of contained in the centralizer of .
Lemma 3.9.
Let and let be an -invariant probability measure on with exponentially small mass in the cusps. Let be a tempered continuous cocycle.
For any amenable subgroup and any Følner sequence of precompact sets in , if the family has uniformly exponentially small mass in the cusps then for any subsequential limit of we have
Proof.
First note that Lemma 3.4 implies the family has uniformly exponentially small mass in the cusps. Note also that for every , the measure is -invariant.
We first claim that for every . For define and let . As is precompact, from Claim 3.6 we have that .
For and , the cocycle property and subadditivity of norms yields
Using that is -invariant, we have for every that
Dividing by yields . The reverse inequality is similar.
The inequality then follows from the upper-semicontinuity in Lemma 3.8. ∎
Consider now any with , a point , and . The empirical measure along the orbit until time is the measure defined as follows: given a bounded continuous , the integral of with respect to the empirical measure is
Similarly, given a probability measure on , the empirical distribution of along the orbit of until time is defined as
Consider now sequences with and . For part (c) of the following lemma, we add an additional assumption that the action of on has uniform displacement: for any compact there is such that for all and ,
Lemma 3.10.
Suppose the action of on has uniform displacement and let be a tempered continuous cocycle.
Let and be sequences with for all and . Let be a sequence of Borel probability measures on and define to be the empirical distribution of along the orbit of for . Assume that
-
(1)
the family of empirical distributions defined above has uniformly exponentially small mass in the cusps; and
-
(2)
.
Then
-
(a)
the family is pre-compact;
-
(b)
for any subsequential limit any subsequential limit of is invariant under the 1-parameter subgroup ;
-
(c)
.
Proof of Lemma 3.10 (a) and (b).
As in the proof of Lemma 3.8, from the assumption that has uniformly exponentially small mass in the cusps we obtain uniform bounds
for all . Combined with the properness of , this establishes uniform tightness of the family of measures and (a) follows.
For (b), let be a compactly supported continuous function. Then for any
The first integral converges to zero as the functions converges uniformly to zero in for fixed . The second integral clearly converges to zero since for we have
which converges to 0 as as is bounded. ∎
The proof of Lemma 3.10(c) is quite involved. It is the analogue in the non-compact setting of [BFH, Lemma 3.6]; we recommend the reader read the proof of of [BFH, Lemma 3.6] first. Two technical complications arise in the proof of Lemma 3.10(c). First, we must control for “escape of Lyapunov exponent” as our cocycle is unbounded. Second, in [BFH] it was sufficient to consider the average of Dirac masses along a single orbit ; here we average measures along an orbit of
To prove Lemma 3.10(c) we first introduce a number of standard auxiliary objects. Let denote the projectivization of the tangent bundle . We represent a point in as where is an equivalence class of non-zero vectors in the fiber . For each , let be a nowhere vanishing Borel section such that
for every . The -action on by vector-bundle automorphisms induces a natural -action on which restricts to projective transformations between each fiber and its image. For each , let be the probability measure on given as follows: given a bounded continuous define
We have that projects to under the natural projection ; moreover, if is a subsequence converging to then any weak- subsequential limit of projects to .
Define by
Note for each fixed that satisfies a cocycle property:
| (13) |
By hypothesis, there are , , and such that
for all and
for all and with
For each , let
As we assume the -action on has uniform displacement, take
We have
If then for every there is an interval of length on which
for all It follows that
By Jensen’s inequality we have
whence
Since we have
| (14) | ||||
In particular, we have
Since
goes to 0 as it follows that
| (15) | ||||
With the above objects and estimates we complete the proof of Lemma 3.10.
Proof of Lemma 3.10 (c).
Consider first the expression We have
Note that the contribution of the second integral is bounded by
which goes to zero as .
Repeatedly applying the cocycle property (13) of we have for that
From (14), the contribution of the second and third integrals is bounded by
which tend to zero as . We then conclude from (15) that
| (16) |
As the family
has uniformly exponentially small mass in the cusps we have
and hence for all . It follows for all that—letting denote the image of in —we have for any with that
In particular, given any , by taking sufficiently large we may ensure that
for any
Since the restriction of to is compactly supported, it is uniformly continuous whence
as In particular given we may take and sufficiently large so that
Let . Note for each that
It then follows for any
where the third equality follows from the invariance of and the cocycle property of . Since
we conclude that
for any whence the result follows. ∎
3.7. Oseledec’s theorem for cocycles over actions by higher-rank abelian groups
Let be a split Cartan subgroup. Then where is the rank of . We have the following consequence of the higher-rank Oseledec’s multiplicative ergodic theorem (c.f. [BRH, Theorem 2.4]).
Fix any norm on and let be
Proposition 3.11.
Let be an ergodic, -invariant Borel probability measure on and suppose . Then there are
-
(1)
an -invariant subset with ;
-
(2)
linear functionals for ;
-
(3)
and splittings into families of mutually transverse, -measurable subbundles defined for
such that
-
(a)
and
-
(b)
for all and all .
Note that (b) implies for the weaker result that for ,
Also note that for , and an -invariant, -ergodic measure that
| (17) |
If is not -ergodic, we have the following.
Claim 3.12.
Let be an -invariant measure with and for some . Then there is an -ergodic component of with
-
(1)
;
-
(2)
there is non-zero Lyapunov exponent for the -action on
We have the following which follows from the above definitions.
Lemma 3.13.
Let be an -invariant probability measure on with exponentially small mass in the cusps. Suppose that is a tempered cocycle. Then for all . In particular, .
3.8. Applications to the suspension action
We summarize the previous discussion in the setting in which we will apply the above results in the sequel. Recall we work with in a fiber bundle with compact fiber
over non-compact base . From the discussion in [BRHW, Section 2.1], we may equip with a metric that is
-
(1)
-invariant;
-
(2)
the restriction to -orbits coincides with the fixed right-invariant metric on ;
-
(3)
there is a Siegel fundamental set on which the restrictions to the fibers of the metrics are uniformly comparable.
The metric then descends to a Riemannian metric on . We fix this metric for the remainder. It follows that the diameter of any fiber of is uniformly bounded. It then follows that if is a measure on then the image in has exponentially small mass in the cusps if and only if does; moreover, a family of probability measures on has uniformly exponentially small mass in the cusps if and only if the family of projected measures on does. Note that by averaging the metric over the left-action of , we may also assume that the metric is left--invariant. This, in particular, implies the right-invariant metric on in above is left--invariant.
For the remainder, the cocycle of interest will be the fiberwise derivative cocycle on the fiberwise tangent bundle,
Given and a -invariant probability measure on , the average leading Lyapunov exponent for the fiberwise derivative cocycle for translation by is written either as or as .
The next observation we need is a variant of a fairly standard observation about cocycle over the suspension action.
Lemma 3.14.
The fiberwise derivative cocycle is tempered.
Proof.
Write . By the construction of the metric in the fibers of there is a with the following properties: given and , writing we have
and
The conclusion is then an immediate consequence of Lemma 2.1. ∎
We now assemble the consequences of the results in this section in the form we will use them below in a pair of lemmas. The first is just a special case of Corollary 3.7.
Lemma 3.15.
Let and let be an -invariant measure on with exponentially small mass in the cusps. Let be an -invariant measure on projecting to . Then the average leading Lyapunov exponent for the fiberwise derivative cocycle, is finite.
The second lemma summarizes the above abstract results in the setting of acting on .
Lemma 3.16.
Let and let be an -invariant measure on with exponentially small mass in the cusps. Let be an -invariant measure on projecting to .
-
(1)
For any amenable subgroup , if is -invariant then
-
(a)
for any Følner sequence of precompact sets in , the family has uniformly exponentially small mass in the cusps; and
-
(b)
for any subsequential limit of we have
-
(a)
-
(2)
For any one-parameter unipotent subgroup centralized by
-
(a)
the family has uniformly exponentially small mass in the cusps; and
-
(b)
for any accumulation point of as we have
-
(a)
Proof.
We remark that we will also use Lemma 3.10 in the proof of the main theorem, but we do not reformulate a special case of it here since the reformulation adds little clarity.
4. Subexponential growth of derivatives for unipotent elements
In this section we show that the restriction of the action to certain unipotent elements in each copy have uniform subexponential growth of derivatives with respect to a right-invariant distance on . Note that each is geodesically embedded whence the distance is the same as the distance. By [LMR1, LMR2], the distance is quasi-isometric to the word-length in . Recall that denotes a right-invariant distance on and that is the identity in .
For , let be the copy of in corresponding to the elements in which acts only on the lattice generated by . Note that as all are conjugate under the Weyl group, it suffices to work with one of them.
Define the unipotent element viewed as an element of . Note that any upper or lower triangular unipotent element of is conjugate to a power of under the Weyl group.
Proposition 4.1 (Subexponential growth of derivatives for unipotent elements).
For any and any , there exists such that for any :
To establish Proposition 4.1, we first show that generic elements in have uniform subexponential growth of derivatives. This first part requires reusing most of the key arguments from [BFH] in a slightly modified form. We encourage the reader to read that paper first.
4.1. Slow growth for “most” elements in
For , , and , we make the following definitions:
-
(1)
For let denote the Haar-volume of .
-
(2)
Let . For let denote the Haar-volume of .
-
(3)
Let denote the ball of radius centered at in .
-
(4)
Let . Given write for the cardinality of .
-
(5)
Define the set of -bad elements to be
-
(6)
Define the set of -good elements to be
To establish Proposition 4.1, we first show that the set contains a positive proportion of when is large enough.
Proposition 4.2.
For any , the set has at least elements for every sufficiently large .
We have the following well-known fact. See for instance [EM1, Section 2].
Lemma 4.3.
There exist positive constants such that for any :
For an element , let denote the projection in . Define
Let
Lemma 4.4.
For almost every and any we have
for all sufficiently large.
Proof.
Let be the matrix
Recall that the action of the one-parameter diagonal subgroup on is ergodic with respect to Haar measure.
Let denote the set of Borel probability measures on equipped with the standard topology (dual to bounded continuous functions). The topology on is metrizable (see [Bil, Theorem 6.8]); fix a metric on on .
Consider the function given by where is the identity coset and is chosen sufficiently small so that is with respect to the Haar measure. By the pointwise ergodic theorem, for almost every and almost every we have
| (18) |
Similarly, for almost every and almost every we have
| (19) |
Let be the set of such that (18) and (19) hold for almost every . The set is -invariant and co-null. We show any satisfies the conclusion of the lemma.
For fixed and fixed , there exist , a sequence for , and a set such that with the property that for any and any we have
| (20) |
and for each
| (21) |
for all . To finish the proof of the lemma, define the set
For large enough, we have that . We claim that
| (22) |
for sufficiently large. For the sake of contradiction, suppose (22) fails. Using that the norm on is chosen to be -invariant, there exists with each in the -orbit of such that for some sequence . Moreover, the corresponding empirical measures
have uniformly exponentially small mass in the cusps by equation (20).
By Lemma 3.10 and (21), a subsequence of the measures converge to an -invariant measure on whose projection to is Haar measure on the embedded modular surface and has positive fiberwise Lyapunov exponent for the action of . Since is ergodic on , we can assume is ergodic by taking an ergodic component without changing any other properties.
We average as in [BFH] to improve to a measure whose projection is the Haar measure on . Difficulties related to escape of mass are handled by the preliminaries in Section 3.
As above, we note that there is a canonical copy of in commuting with our chosen . Recall is the Cartan subgroup of of positive diagonal matrices. The subgroup contains the one-parameter group and a Cartan subgroup of . Let
-
•
,
-
•
, and
-
•
.
Note that has codimension one. Our chosen modular surface is such that
Define an -ergodic, -invariant measure on that projects to Haar measure on as follows: Let denote the restriction of the fiber-bundle to . Pick point in that equidistributes to the Haar measure on under a Følner sequence in . Consider as a measure on the restriction of to . Now average over a Følner sequence in and take a limit . Note that has positive fiberwise Lyapunov exponent . This can be seen by mimicking the proof of Lemma 3.9. Let be an ergodic component of , then the measure has the desired properties and is supported on the subset of defined by restricting the bundle to .
We consider the -action on and the fiberwise derivative cocycle . By (17), there is a non-zero Lyapunov exponent for this action. We apply the averaging procedure in Proposition 3.5 to this measure. Take to be either or so that is not proportional to . Choose such that and . Let and let be any subsequential limit of as . Then is -invariant, and has positive fiberwise Lyapunov exponent . Moreover, is -invariant. By Lemma 3.16 and Proposition 3.5, has exponentially small mass in the cusps. We may also assume is ergodic by passing to an ergodic component and by Claim 3.12 assume has a non-zero fiberwise Lyapunov exponent for the -action.
We now average over to obtain . Then has a non-zero fiberwise Lyapunov exponent and has exponentially small mass in the cusps by Lemma 3.16(1). Since was -invariant, we have . Once again, we may pass to an -ergodic component of that retains the desired properties.
Take to be either or so that is not proportional to on . Select with and . By Proposition 3.5 and Lemma 3.16, we obtain a new measure with the Haar measure on . We have . Finally, average over all of to obtain . Since is the Haar measure and thus -invariant, we have that . By Lemma 3.16, has a non-zero fiberwise Lyapunov exponent for the action of . Replace by an ergodic component with positive fiberwise Lyapunov exponent.
Exactly as in [BFH, Section 5.5], we apply [BRHW, Proposition 5.1] and conclude that is a -invariant measure on . We then obtain a contradiction with Zimmer’s cocycle superrigidity theorem. To conclude that is a -invariant, note that [BRHW, Proposition 5.1] holds for actions induced from actions of any lattice in and shows that is invariant under root subgroups corresponding to non-resonant roots. Dimension counting exactly as in [BFH, Section 5.5] shows that the non-resonant roots of generate all of if the dimension of is at most or if the dimension of is and the action is preserves a volume. ∎
Proof of Proposition 4.2.
Fix sufficiently small so that if then . Fix a point as in Lemma 4.4 with . Observe that if and , then . In particular, for any we have for all sufficiently large that
| (23) |
where is a constant depending on .
Take to be the ball of radius centered at the identity coset in and consider lifts of to intersecting the ball . If a lift of intersects , then the corresponding element of the deck group belongs to .
4.2. Subexponential growth of derivatives for unipotent elements in
We work here with a specific copy of the group embedded in and its intersection with the lattice ; the copy of corresponds to the elements of which differ from the identity matrix only in the first two rows and first three columns. Any unipotent element of any considered in the statement of Proposition 4.1 is conjugate by an element of the Weyl group to a power of the elementary matrix . Thus, after conjugation, any such element is contained in the distinguished copy of generated by and the normal subgroup generated by and .
For the reminder of this subsection, we work with this fixed group. Identify with . Let denote the abelian subgroup of consisting of unipotent elements of the form
Clearly, is normalized by and . We have an embedding
where is identified with the subgroup generated by the unipotent elements and . Note that is a torus bundle over the unit-tangent bundle of the modular surface.
Equip with the norm with respect to the generating set and let denote the closed ball of radius in centered at with respect to this norm. Given let denote the cardinality of the set .
Define the set of “-good unipotent elements” of , denoted by , to be the following subset of :
| (24) |
The main results of this subsection is the following.
Proposition 4.6.
For any , there exists such that if , then
Proposition 4.1 follows from Proposition 4.6 using that any subgroup in Proposition 4.1 is conjugate to a subgroup of the group and the fact that from (6). The proof of Proposition 4.6 consists of conjugating elements of by elements of in order to obtain a subset of that contains a positive density of elements of . Then, using the fact that is abelian, we promote such a subset to all of by taking sufficiently large sumsets in Proposition 4.9.
Lemma 4.7.
There exists with the following properties: for any there is an such that for any we have
Proof.
Recall that denotes the intersection of the ball of radius in with and denotes the cardinality of . As grows exponentially in , we may take fixed so that for all sufficiently large. Given , define the subset to be
From Proposition 4.2, we may assume that
From (3), there exists such that if belongs to then either
| or . |
Without loss of generality, we assume that at least half of the elements in satisfy .
Consider the map that assigns to . By (3), there is such that the image of lies in the norm-ball for all .
Let . Then . If is sufficiently large and then we have ; indeed
whence
We have for some . Also, from (4) and Lemma 4.3 we have for some .
To to complete the proof, we show that the preimage in of any satisfying has uniformly bounded cardinality depending only on . Observe that if satisfy , then , where for some and we have
If belongs to then and if belongs to then . We thus have that and . As we assume that
we have that . Thus, the preimage has at most elements in .
With , having taken sufficiently large, we thus have
which completes the proof.∎
To complete the proof of Proposition 4.6, we show that any element in can be written as a product of a bounded number of elements in independent of . This follows from the structure of sumsets of abelian groups.
From the chain rule and submultiplicativity of norms, we have the following.
Claim 4.8.
For any positive integers and , if and then the product
For subsets we denote by the sumset of .
Claim 4.9.
For any , there exists a positive integer and a finite set such that for any and any symmetric set with , we have that
Proof.
Fix with . Take , , and Consider a symmetric set with .
If then and we are done. Thus, consider . To complete the proof the claim, we argue that the set
contains the intersection of the sublattice with . Adding to the sumset then implies the claim. Consider any non-zero vector of the form for some . Then where is such that .
Consider the equivalence relation in defined by declaring that two elements are equivalent if is an integer multiple of . Each equivalence class is of the form
As , there exists one equivalence class such that . Since , each equivalence class contains at least elements and hence contains at least two elements with for . In particular, since , we have . As divides , we have that .
Similarly, for and any of the form we have . Then
completing the proof. ∎
Proof of Proposition 4.6.
Given , let and be given by Lemma 4.7. Let be as in (24) and take and as in Lemma 4.9. Note that is symmetric by definition. Take such that whenever . For and any we have that ( times) by Proposition 4.9. Proposition 4.8 then implies that so . With , take Then for all we have
whence
and for with . ∎
5. Proof of Theorem B
5.1. Reduction to the restriction of an action by
We recall the work of Lubotzky, Mozes, and Raghunathan, namely [LMR1] and [LMR2], which establishes quasi-isometry between the word and Riemannian metrics on lattices in higher-rank semisimple Lie groups. In the special case of for , in [LMR1, Corollary 3] it is shown that any element of is written as a product of at most elements . Moreover each is contained some and the word-length of each is proportional to the word-length of .
Thus, to establish that an action has uniform subexponential growth of derivatives in Theorem B, it is sufficient to show that the restriction has uniform subexponential growth of derivatives for each . We emphasize that to measure subexponential growth of derivatives, the word-length on is measured as the word-length as embedded in (which is quasi-isometric to the Riemannian metric on ) rather than the intrinsic word-length in (which is not quasi-isometric to the Riemannian metric on ).
As the Weyl group acts transitively on the set of all , it is sufficient to consider a fixed . Thus to deduce Theorem B, in the remainder of this section we establish the following, which is the main proposition of the paper.
Proposition 5.1.
For any action as in Theorem B, the restricted action has uniform subexponential growth of derivatives.
5.2. Orbits with large fiber growth yet low depth in the cusp
To prove Proposition 5.1, as in Section 4.2 we consider a canonical embedding of in . Write
for the geodesic flow on . Let be a fixed compact -invariant “thick part” of ; that is, relative to the Dirichlet domain in (7), points in corresponds to the points in whose imaginary part is bounded above, say, by .
A geodesic curve in the modular surface of length corresponds to the image of an orbit where and . Denote the length of such a curve by . For an orbit of in we define
The following claim is straightforward from the compactness and the quasi-isometry between the word and Riemannian metrics on .
Claim 5.2.
For an action , the following statements are equivalent:
-
(1)
the restriction has uniform subexponential growth of derivatives;
-
(2)
for any there is a such that for any orbit with , , and we have
Define the maximal fiberwise growth rate of orbits starting and returning to to be
| (25) |
Using Claim 5.2, to establish Proposition 5.1 it is sufficient to show that .
For an orbit , define the following function which measures the depth of into the cusp:
The following lemma is the main result of this subsection.
Lemma 5.3.
If then there exists a sequence of orbits with , , and such that
-
(1)
;
-
(2)
.
We first have the following claim.
Claim 5.4.
For any there exists with the following properties: for any and such that for all and then, for the orbit , we have
Indeed, the claim follows from the fact that the value of the return cocycle is defined by geodesic in the cusp of is given by a unipotent matrix of the form and Proposition 4.1.
Proof of Lemma 5.3.
Let be a sequence of orbits with , , , and such that
Replacing with a subsequence, we may assume the following limit exists:
We aim to prove that . Arguing by contradiction, suppose . We decompose the orbit
as a concatenation of smaller orbit segments with the following properties:
-
(1)
each orbit is such that ;
-
(2)
the endpoints of each orbit are contained in ;
-
(3)
each orbit is contained entirely in with endpoints contained in ;
-
(4)
each orbit satisfies whence for sufficiently large.
Note for each , that and thus is bounded by some independent of . Additionally, since is finitely generated and (equipped with the word metric) is quasi-isometrically embedded in , there exists a constant such that for any orbit segment whose endpoints are contained in , we have . By the definition of , for any there is a positive constant such that for any orbit sub-segment
-
(1)
whenever
-
(2)
whenever .
From Claim 5.4, for any we have, assuming that and hence are sufficiently large, that
for all orbit sub-segmants .
5.3. Construction of a Følner sequence and family averaged measures
Assuming that in (25) is non-zero, we start from the orbit segments constructed in Lemma 5.3 and perform an averaging procedure to obtain a family of measures on whose properties lead to a contradiction. In particular, the projection of any weak- limit of to will be -invariant, well behaved at the cusps, and have non-zero Lyapunov exponents. These measures on are obtained by averaging certain Dirac measures against Følner sequences in a certain amenable subgroup of .
Consider the copy of as the subgroup of matrices that differ from the identity away from the th row and th column. Let be the abelian subgroup of unipotent elements that differ from the identity only in the th column; that is given a vector define to be the unipotent element
| (29) |
and let . is normalized by .
Identifying with we have an embedding . The subgroup has as a lattice the subgroup
and there is a natural embedding given by the inclusion
Recall is the group of diagonal matrices with positive entries. Let denote matrices
Complete the set to a spanning set of viewed as vector space where the are diagonal matrices whose -entry is equal to .
Let be the subset of consisting of all the elements of the form
| (30) |
where, for some to be determined later (in the proof of Proposition 5.10 below),
-
(1)
;
-
(2)
;
-
(3)
;
-
(4)
.
Claim 5.5.
is Følner sequence in .
Observe that is linearly-long in the -direction and exponentially-long in the -direction. From conditions (2) and (4), the -component of is much longer in the -direction than in the other directions. The condition (2) that is fundamental in our estimates in Section 5.4 that ensure the measures constructed below have uniformly exponentially small mass in the cusps. These estimates are related to the fact that orbits of correspond to the unstable manifolds for the flow defined by in and open subsets of unstable manifolds equidistribute to the Haar measure on under the flow .
Recall we have a sequence of fiber bundles
and may consider as a fiber bundle over . Given , let denote the fiber of over . An element is a pair where, identifying the fiber of through with , we have and . Given , we write using our chosen norm on . Given , let denote the footpoint of in the fiber of through .
If uniform subexponential growth of derivatives fails for the restriction of the to , then there exist sequences , with , and as in Lemma 5.3 and Claim 5.2 with , such that
| (31) |
for some .
Note that is a solvable group. We may equip with any left-invariant Haar measure. Note that the ambient Riemannian metric induces a right-invariant Haar measure on but as is not unimodular these measures do not coincide.
For each , take to be the measure on obtained by averaging the Dirac measure over the set :
where is the volume of and indicates integration with respect to left-invariant Haar measure on .
We expand the above integral in our coordinates introduced above. Then for any bounded continuous function , integrating against our Euclidean parameters and we have
| (32) | ||||
where denotes the volume of
with respect to the Euclidean parameters .
For each , let denote the image of the measure under the canonical projection from to . The following proposition is shown in the next subsection.
Proposition 5.6.
There exists such that the sequence of measures has uniformly exponentially small mass in the cusp with exponent .
By the uniform comparability of distances in fibers of , this implies the family of measures has uniformly exponentially small measure in the cusp.
By Lemma 3.10(a) the families of measures and are precompact families. As is a Følner sequence in a solvable group, we have that any weak- subsequential limit of or is -invariant. Moreover, from Theorem 3.1(d), it follows that any weak- subsequential limit of is invariant under the group generated by the root groups for each . Since and generate all of , we have that is a -invariant measure on .
5.4. Proof of Proposition 5.6
5.4.1. Heuristics of the proof
The heuristic of the proof is the following. Observe that for a fixed choice of and as given by the choice of Følner set , the point
lies at sub-linear distance to the thick part of with respect to . Observe that the -orbit of such point is an embedded -dimensional torus in . As the range of points in in the Følner set is quite large, averaging a Dirac measure of the point in the -direction in yields a measure quite close to Haar measure on the -orbit.
Observe that -orbits correspond to unstable manifolds for the action of the flow on . As the action of is known to be mixing, we expect that if is sufficiently large, flowing by the -orbit of will become equidistributed and in particular it will intersect non-trivially the thick part of . This is the reason why the condition is assumed.
While intuition about mixing motivates the proof, we do not use it explicitly. Instead we use that for large enough , the action of expands the -orbits in a way that forces them to hit the thick part. We verify this fact by explicit matrix multiplication.
As normalizes , the image under of the -orbit of is the -orbit of a point in the thick part of . Having in mind the quantitative non-divergence of unipotent flows as in the proof Proposition 3.2, the -orbits have uniformly (over all and ) exponentially small mass in the cusps whence so do the measures .
The following proof of Proposition 5.6 uses explicit matrix calculations and estimates to verify these heuristics.
5.4.2. Proof of Proposition 5.6
Recall that we identify each coset
with a unimodular lattice in . We define the systole of a unimodular lattice to be
and for an element , we denote by the systole
From (9), to prove Proposition 5.6 it is sufficient to find so that the integrals
are uniformly bounded in .
As discussed in the above heuristic, from (32) to bound the integrals it is sufficient to show each integral
is uniformly bounded in and in all parameters for and Recall here that are the points satisfying (31) used in the construction of the measures .
We have is canonically embedded in . Given , let
denote the element mapping to under the map which is contained in a fundamental domain contained in the Dirichlet domain in (7) in Section 2.2. Let denote the operator norm on and the associated conorm.
Claim 5.7.
For every , , and as above, there exist
| and |
such that:
-
(1)
-
(2)
and
Proof.
In the remainder, we will suppress the dependence of choices on . We take be such that
Note that differs from the identity only in the first two rows and columns. Since each is contained in , we have that the matrix norm and conorm and are bounded above and below, respectively, by constants and independent of .
Recall denotes a vector in and is the unipotent element given by (29). Matrix computation yields
whence
We have
| (33) |
To reduce notation, for fixed and define
We aim to find an upper bound of
that is independent of and and .
Observe that if differ by an element of the unimodular lattice , then . Indeed, if for some and if is then
where Thus we have that descends to a function on the torus .
Let be a fundamental domain for this torus in centered at . Let denote the number of -translates of that intersect . Then, if is sufficiently large we have that
The first inequality follows from inclusion. The second inequality follows from the fact that the perimeter of grows like , the volume of grows like , and the domains have uniformly comparable geometry over .
It remains to estimate . Given and fixed and we define
Proposition 5.6 follows immediately from the estimate in the following lemma.
Lemma 5.8.
There exists constants , independent of and , such that
Proof of Lemma 5.8.
From (33), given any , if then there exists a non-zero such that
which (as ) holds if and only if there is a non-zero
| (34) |
As induces a volume-preserving automorphism of , the set of satisfying (34) for some has the same measure as the set of satisfying
for some
For every integer satisfying , let be the subset of such that there exists satisfying
Then Thus the estimate reduces to the following.
Claim 5.9.
There exists such that for all sufficiently large.
Proof.
Recall that . If then, for any non-zero , we have
From Claim 5.7(2), if is large enough then and so the term in the left hand side above is greater than one, therefore for sufficiently large.
If , observe that the map given by
preserves the Lebesgue measure on In particular, this implies that and have the same volume.
We thus take . Note that . There is a , depending only on , such that the set
has cardinality at most . From Claim 5.7(2), if is large enough then whence for all and all ,
We thus need only consider .
Given a fixed , using that we have
whence
If so that
then
| (35) |
Since the set of satisfying (35) has the same volume as the set of satisfying
It follows that .∎
5.5. Positive Lyapunov exponents for limit measures
To deduce Proposition 5.1, having assumed that in (25) is non-zero, we show that any weak- subsequential limit of the sequence of measures has a positive Lyapunov exponent from which we derive a contradiction.
Recall from Section 5.3 that we fixed sequences such that for some fixed . Let be the fiberwise derivative cocycle over the action of on .
Our main result is the following.
Proposition 5.10.
For any weak- subsequential limit of we have
We first show that averaging over does not change the Lyapunov exponents of the cocycle.
Claim 5.11.
Given any there is such that for any and any we have
for any .
Proof.
Recall that the -orbit of any is a closed torus. Then the -orbit of is compact. Recall our fixed fundamental domain contained in the Dirichlet domain of the identity for as discussed in Section 2.2. Given , let be the lift of in . Let denote the lift of to and let be the lift of the orbit to . As discussed in Section 2.2, we have that is contained in the Dirichlet domain of the identity for the -action on . Moreover, is precompact in .
Fix and . Write
for some ; we have and for all . The deck group of the orbit is
Thus, there is and such that
and . Then
Since and are in precompact sets, the first and last terms of the right hand side are uniformly bounded in and .
There exists some such that
Since we have whence for some constants and . Proposition 4.1 implies for any that
and taking sufficiently small, the claim follows. ∎
By Lemma 2.1, the fact that is finitely generated, and the uniform comparability of the fibers of , we also have the following.
Claim 5.12.
There are uniform constants and with the following property: Let . Then for any with we have
We now prove Proposition 5.10.
Proof of Proposition 5.10.
Recall we take , , and with such that for some fixed in (31) in Section 5.3. We also write for the fiberwise derivative cocycle.
The measures constructed in Section 5.3 are defined by averaging last along the orbit . Let be the measure on given by
In the context of Lemma 3.10, the measures constructed in Section 5.3 correspond to the empirical measures appearing in the proof of Lemma 3.10. From Lemma 3.10, to establish Proposition 5.10 it is sufficient to show that
We have
Consider fixed and . Take such that . Then
Observe that both and are contained in a fixed compact subset of and hence, by Claim 5.12, having taken sufficiently small in the construction of the Følner sequence, from the constraints on and we have and for all sufficiently large.
Moreover, from Claim 5.11, we have for all sufficiently large.
5.6. Proof of Proposition 5.1
Having assumed that in (25) is non-zero, we arrive at a contradiction. Take any weak- subsequential limit of the sequence of measure on . We have that is -invariant and has a non-zero fiberwise Lyapunov exponent for the fiberwise derivative over the action of . Moreover, we have that projects to on which, as discussed above, is the Haar measure on . We may replace with an -ergodic component with the same properties as above. Then is -ergodic, projects to Haar, and the fiberwise derivative cocycle over the -action on has a non-zero Lyapunov exponent functional
As in the conclusion of Lemma 4.4, the arguments of [BFH, Section 5.5] using [BRHW, Proposition 5.1] imply that the measure is, in fact, -invariant. As before, we note that [BRHW, Proposition 5.1] does not assume is cocompact, so the algebraic argument applying that proposition in [BFH, Section 5.5] goes through verbatim. For a more self-contained proof that applies since we only consider the case of see [BDZ, Proposition 4]. We then obtain a contradiction with Zimmer’s cocycle superrigidity by constraints on the dimension of the fibers of . Thus we must have and Proposition 5.1 follows.
References
- [Ath] J. S. Athreya. Quantitative recurrence and large deviations for Teichmuller geodesic flow, Geom. Dedicata 119(2006), 121–140.
- [Bil] P. Billingsley. Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons, Inc., New York, second edition, 1999. A Wiley-Interscience Publication.
- [BDZ] A. Brown, D. Damjanovic, and Z. Zhang. actions on manifolds by lattices in Lie groups. Preprint (2018). arXiv:1801.04009.
- [BFH] A. Brown, D. Fisher, and S. Hurtado. Zimmer’s conjecture: Subexponential growth, measure rigidity, and strong property (T), Preprint (2016). arXiv:1608.04995.
- [BRH] A. Brown and F. Rodriguez Hertz. Smooth ergodic theory of -actions part 1: Lyapunov exponents, dynamical charts, and coarse Lyapunov manifolds. Preprint (2016). arXiv:1610.09997.
- [BRHW] A. Brown, F. Rodriguez Hertz, and Z. Wang. Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds. Preprint (2016). arXiv:1609.05565.
- [BM] M. Burger and N. Monod. Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal. 12(2002), 219–280.
- [dlS] M. de la Salle. Strong (T) for higher rank lattices. (2017). arXiv:1711.01900.
- [EM1] A. Eskin and C. McMullen. Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71(1993), 181–209.
- [EM2] A. Eskin and M. Mirzakhani. Counting closed geodesics in moduli space, J. Mod. Dyn. 5(2011), 71–105.
- [FM] D. Fisher and G. A. Margulis. Local rigidity for cocycles. In Surveys in differential geometry, Vol. VIII (Boston, MA, 2002), volume 8 of Surv. Differ. Geom., pages 191–234. Int. Press, Somerville, MA, 2003.
- [FH] J. Franks and M. Handel. Distortion elements in group actions on surfaces, Duke Math. J. 131(2006), 441–468.
- [Ghy] É. Ghys. Actions de réseaux sur le cercle, Invent. Math. 137(1999), 199–231.
- [KKLM] S. Kadyrov, D. Y. Kleinbock, E. Lindenstrauss, and G. A. Margulis. Singular systems of linear forms and non-escape of mass in the space of lattices. Preprint (2016). arXiv:1407.5310.
- [KM] D. Y. Kleinbock and G. A. Margulis. Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2) 148(1998), 339–360.
- [Kle] D. Kleinbock. An extension of quantitative nondivergence and applications to Diophantine exponents, Trans. Amer. Math. Soc. 360(2008), 6497–6523.
- [LMR1] A. Lubotzky, S. Mozes, and M. S. Raghunathan. Cyclic subgroups of exponential growth and metrics on discrete groups, C. R. Acad. Sci. Paris Sér. I Math. 317(1993), 735–740.
- [LMR2] A. Lubotzky, S. Mozes, and M. S. Raghunathan. The word and Riemannian metrics on lattices of semisimple groups, Inst. Hautes Études Sci. Publ. Math. (2000), 5–53 (2001).
- [Mar1] G. A. Margulis. Explicit constructions of expanders, Problemy Peredači Informacii 9(1973), 71–80.
- [Mar2] G. Margulis. Problems and conjectures in rigidity theory. In Mathematics: frontiers and perspectives, pages 161–174. Amer. Math. Soc., Providence, RI, 2000.
- [Pol] L. Polterovich. Growth of maps, distortion in groups and symplectic geometry, Invent. Math. 150(2002), 655–686.
- [Rat1] M. Ratner. Invariant measures and orbit closures for unipotent actions on homogeneous spaces, Geom. Funct. Anal. 4(1994), 236–257.
- [Rat2] M. Ratner. On Raghunathan’s measure conjecture, Ann. of Math. (2) 134(1991), 545–607.
- [Sha1] N. A. Shah. Limit distributions of polynomial trajectories on homogeneous spaces, Duke Math. J. 75(1994), 711–732.
- [Sha2] Y. Shalom. Rigidity of commensurators and irreducible lattices, Invent. Math. 141(2000), 1–54.
- [Wit] D. Witte. Arithmetic groups of higher -rank cannot act on -manifolds, Proc. Amer. Math. Soc. 122(1994), 333–340.